Integrand size = 7, antiderivative size = 21 \[ \int \cos (x) \csc (3 x) \, dx=\frac {1}{3} \log (\sin (x))-\frac {1}{6} \log \left (3-4 \sin ^2(x)\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4441, 272, 36, 31, 29} \[ \int \cos (x) \csc (3 x) \, dx=\frac {1}{3} \log (\sin (x))-\frac {1}{6} \log \left (3-4 \sin ^2(x)\right ) \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 272
Rule 4441
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \left (3-4 x^2\right )} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(3-4 x) x} \, dx,x,\sin ^2(x)\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\sin ^2(x)\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{3-4 x} \, dx,x,\sin ^2(x)\right ) \\ & = \frac {1}{3} \log (\sin (x))-\frac {1}{6} \log \left (3-4 \sin ^2(x)\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \cos (x) \csc (3 x) \, dx=\frac {1}{3} \log (\sin (x))-\frac {1}{6} \log \left (3-4 \sin ^2(x)\right ) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.68 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{3}-\frac {\ln \left ({\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}+1\right )}{6}\) | \(27\) |
default | \(-\frac {\ln \left (1+2 \cos \left (x \right )\right )}{6}-\frac {\ln \left (2 \cos \left (x \right )-1\right )}{6}+\frac {\ln \left (\cos \left (x \right )-1\right )}{6}+\frac {\ln \left (\cos \left (x \right )+1\right )}{6}\) | \(34\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \cos (x) \csc (3 x) \, dx=-\frac {1}{6} \, \log \left (4 \, \cos \left (x\right )^{2} - 1\right ) + \frac {1}{3} \, \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \]
[In]
[Out]
Time = 0.56 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \cos (x) \csc (3 x) \, dx=- \frac {\log {\left (4 \sin ^{2}{\left (x \right )} - 3 \right )}}{6} + \frac {\log {\left (\sin {\left (x \right )} \right )}}{3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (17) = 34\).
Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 6.14 \[ \int \cos (x) \csc (3 x) \, dx=-\frac {1}{12} \, \log \left (2 \, {\left (\cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - \frac {1}{12} \, \log \left (-2 \, {\left (\cos \left (x\right ) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + \frac {1}{6} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac {1}{6} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \cos (x) \csc (3 x) \, dx=\frac {1}{6} \, \log \left (-\cos \left (x\right )^{2} + 1\right ) - \frac {1}{6} \, \log \left ({\left | 4 \, \cos \left (x\right )^{2} - 1 \right |}\right ) \]
[In]
[Out]
Time = 26.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \cos (x) \csc (3 x) \, dx=\frac {\ln \left (\sin \left (x\right )\right )}{3}-\frac {\ln \left (\frac {1}{4}-{\cos \left (x\right )}^2\right )}{6} \]
[In]
[Out]